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For basic questions about limits, derivatives, integrals and applications, mainly of one-variable functions.

Calculus, more properly called analysis (or real analysis or in older literature, infinitesimal analysis), is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the length, area and volume of objects. Calculus is sometimes divided into differential and integral calculus, which are concerned with derivatives

\begin{equation*} \frac{dy}{dx}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} \end{equation*}

and integrals

\begin{equation*} \int_a^b f(x)\ dx = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k, \end{equation*}

respectively. These are related through the fundamental theorem of calculus.

While ideas related to calculus had been known for some time (Archimedes' method of exhaustion was a form of calculus), it was not until the independent work of Newton and Leibniz that the modern elegant tools and ideas of calculus were developed. The ideas of integration were later extended by Riemann and Lebesgue. More recently, the Henstock–Kurzweil integral has led to a more satisfactory version of the second part of the Fundamental Theorem of Calculus.

Even so, many years elapsed until the subject was put on a mathematically rigorous footing by mathematicians such as Cauchy and Weierstrass, the latter of whom famously proved an everywhere continuous function can be nowhere differentiable.

Source: Wolfram Mathworld